Round and Round – trig functions

Last week we talked about some basic functions that you will deal with in algebra, calculus and beyond. Today I am going to introduce you to some basic trig functions.

One of the most fascinating applications of trig functions is that of daylight hours. Do you notice how the days get longer in the summer, then shorter in the winter? And then they get longer and shorter and longer and shorter, and the cycle just keeps going on.

Trig functions are just that – cycles.

This is a graph of the function f(x) = sin(x) (the one on the left is one period of the sine function, and the one on the right shows more what the graph does – it just keeps going over and over again – in a cycle.)

 

This is a graph of the function f(x) = cos(x). Cosine is a lot like sine – it just starts in a different place. Where sine starts at zero when x = 0, cosine starts at 1 when x = 0.

This is a graph of the function f(x) = tan(x). Tangent is a ratio of sine and cosine. The reason it is undefined at some places (see how the graph goes up and doesn’t come back down, and then it stars from far below?) is because sometimes cosine is zero, and you can’t divide by zero.

This is just a little introduction to show you what trig functions look like. There are also inverse trig functions, which we’ll talk about later. We’ll also talk about some really interesting applications of trig functions.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Functional Graphs

There are some graphs of basic functions that every math student should know. If you can recognize what a function is going to look like when it is graphed on the xy plane, you’ll be a lot more likely to be able to figure out problems about that function, because you’ll be familiar with is.

Think of this – when someone is talking to you about a person, it’s hard to understand exactly what they are taking about, unless you have some kind of mental picture of the person they are talking about. Otherwise, you’ll be saying “Huh?”

So, to help you avoid saying “huh”, I am going to help you learn to recognize the graphs of basic functions.

First, , the most basic function. And probably the most boring. Blah.

And the next one, , this is a “parabola” – think “bowl.”

 

After the parabola, we must have a “cubic” (think, a “cube” has three dimensions – height, width, length)

Next, , also known as the “absolute value function”.

The slightly odd function, - a tricky function, because it is not defined at x=0!

Now for one of my very favorite functions,(an “exponential” with the famous number “e” – yes, it is a number.)

 

 

And we have have the exponential function without it’s inverse,(which is also not defined at x=0)

So there you have it – the seven basic functions (that are not trig functions – we’ll deal with those in another post). If you can memorize what these look like, you’ll be set through almost all of your math years. But I won’t let you get away with just memorizing them. Eventually we’ll study exactly why these functions look the way they do, and we’ll learn a lot of really amazing stuff about them.

Which is your favorite graph? Why?

(ps – I made these graphs using an amazing FREE graph drawing tool for your computer called WinPlot – if you don’t have a graphing calculator, this will be very useful for you. You can do a lot of the same things you can do on a graphing calculator using this program)

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.